Physically unclonable functions using pulse width chaotic maps

ABSTRACT

Technologies are provided for clockless physically unclonable functions (PUFs) in reconfigurable devices. Embodiments of the disclosed technologies include processing circuitry configured to perform numerous operations. The operations can include receiving a challenge continuous pulse signal, and generating a response continuous pulse signal by iteratively extending the challenge continuous pulse signal in time-domain. In some configurations, the iteratively extending includes generating a next continuous pulse signal by operating on a prior continuous pulse signal according to a stretching function, and generating a second next continuous pulse width signal by operating on the next continuous pulse signal according to a folding function.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No. 16/436,879, filed Jun. 10, 2019, which claims the benefit of U.S. Provisional Patent Application No. 62/682,704, filed Jun. 8, 2018, the contents of which are hereby incorporated by reference in their entireties.

BACKGROUND

Physically Unclonable Functions (PUFs) are physical authentication devices that make use of the microscopic variations in manufactured components. Existing implementations of PUFs commonly use a challenge and response protocol, where a challenge constitutes an input signal and a response constitutes an output signal. The goal any PUF is to exhibit responses that are device dependent and cannot be replicated/predicted through a physical manufacturing process or a software representation of the device. Commonly used PUFs are implemented on logic gate devices and use structures such as ring oscillators, where the frequency of the selected rings oscillators depends on the manufacturing variations in the selected logic gates used in the device and multiple copies of this device thus display unique frequencies. In addition to being device dependent, the behavior of such PUFs must also be difficult to predict via modeling approaches (e.g., machine learning). Conventional PUFs currently lack in first principles theory to ensure their operation across any platform and continue to be susceptible to advances in software based prediction. Thus, current PUFs are inadequate with regard to ensuring broad application across any suitable device via first principles theory and mathematics and are inadequate with regard to being robust against modeling based predicition.

SUMMARY

It is to be understood that both the following general description and the following detailed description are exemplary and explanatory only and are not restrictive. Provided are methods and systems for PUFs using pulse width (i.e., time-based) chaotic maps. The methods and systems operate on an initial pulse width to produce multiple output pulses, where the widths of the output pulses represent the iterations of a chaotic map, for example. The PUF in this description can convert that initial pulse width into a device-dependent digital representation that can be stored or analyzed. Multiple copies of this PUF device can be implemented and compared across different devices. In each device, unique device-dependent digital representation of pulse widths can be produced to construct a database of unique device features. These features can be then used to identify each device at a later time, similar to how a unique fingerprint that can identify an individual. Thus, each pulse width that is mapped to a digital signal (e.g., a binary signal) that can be recorded/represented contains device-dependent information (unique bits) that are produced as a product of the manufacturing variations in the device. The PUF device describe here encodes these manufacturing variations using a pulse width chaotic map (e.g., a tent map, a Bernoulli shift map, a logistic map, or similar), where a pulse width is stretched and folded using time-based operators, where the time-based operators have many unique parameter values that are defined during the devices manufacturing process.

Embodiments of the disclosed technologies can be implemented in fully-electronic assemblies, including transistors; CMOS logic gates; field programmable gate-arrays (FPGAs); application-specific integrated circuits (ASICs); a combination of the foregoing; or similar. Assemblies that include FPGAs are highly practical platform for implementing the principles of this disclosure because, amongst other things, FPGAs are reconfigurable and can be implemented/updated using software only.

In at least some embodiments, an asynchronous digital pulse can have leading edge that corresponds to one signal/event and a trailing edge that corresponds to the next or neighboring signal/event. Thus, the width of this asynchronous digital pulse is an electronic representation of the time interval between these two events. The techniques disclosed herein can addresses the generation of a binary representation of such continuous pulse signals without reliance on a clocking signal.

To do so, real-time pulse operations that transform the duration of a logic signal, resulting in a continuous pulse signal of a desired width in the time domain. In particular, these operations both increase/stretch and decrease/fold the width of a continuous pulse signal according to an iterative chaotic map (e.g., a tent map, a Bernoulli shift map, or similar). This iterative chaotic map also produces binary bits that can be used to represent a digital word. The processes described above can be recursive. For example, the processes can be repeated using an iterative feedback delay loop.

Additional advantages will be set forth in part in the description which follows or may be learned by practice. The advantages will be realized and attained by means of the elements and combinations particularly pointed out in the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The annexed drawings are an integral part of the disclosure and are incorporated into the present specification. The drawings illustrate examples of embodiments of the disclosure and, in conjunction with the description and claims, serve to explain, at least in part, various principles, features, or aspects of the disclosure. Some embodiments of the disclosure are described more fully below with reference to the drawings. However, various aspects and elements of the disclosure can be implemented in many different forms and should not be construed as being limited to the implementations set forth herein. Like numbers refer to like, but not necessarily the same or identical, elements throughout. The accompanying drawings can be briefly characterized as follows.

FIG. 1 illustrates an example of a pulse operation component that is used in the PUF to convert a single input pulse into a continuous-time pulse train of many pulses, where each pulse produces a bit value, in accordance with one or more embodiments of this disclosure.

FIG. 1A illustrates another example of the pulse operations in the PUF via a chaotic map that convert continuous pulse width to a pulse train and corresponding binary bits, in accordance with one or more embodiments of this disclosure.

FIG. 1B illustrates an example of a component that implements a chaotic map, in accordance with one or more embodiments of this disclosure.

FIG. 1C illustrates an example of a component included in the chaotic map PUF that uses multiple chaotic map components in series, where each component operates on a continuous pulse signal, in accordance with one or more embodiments of this disclosure.

FIG. 1D illustrates an example of latch circuitry and timing, in accordance with one or more embodiments of this disclosure.

FIG. 1E illustrates an example of circuitry that constitute a component that implement a pulse-width folding operation, in accordance with one or more embodiments of this disclosure.

FIG. 1F illustrates another example of circuitry that constitute a component that implements a pulse-width folding operation, in accordance with one or more embodiments of this disclosure.

FIG. 1G illustrates an example of circuitry that constitutes a component that implements pulse-width stretching operations, in accordance with one or more embodiments of this disclosure.

FIG. 2 illustrates another example of circuitry that constitutes a component that implements a pulse width stretching operation, in accordance with one or more embodiments of this disclosure.

FIG. 3A illustrates an example of a physically unclonable function (PUF) device, in accordance with one or more embodiments of this disclosure.

FIG. 3B illustrates an example of another PUF device, in accordance with one or more embodiments of this disclosure.

FIG. 3C illustrates an example of yet another PUF device, in accordance with one or more embodiments of this disclosure.

FIG. 4A illustrates an example of pulse operation component that utilizes a time delay component to iteratively operate on a precursor pulse, in accordance with one or more embodiments of this disclosure.

FIG. 4B illustrates an example of another pulse operation component that utilizes a time delay component to iteratively operate on a precursor pulse, in accordance with one or more embodiments of this disclosure.

FIG. 5A illustrates an example of pulse operation component that utilizes multiple time-based chaotic map components assembled in series to iteratively operate on a precursor pulse, in accordance with one or more embodiments of this disclosure.

FIG. 5B illustrates an example of another pulse operation component that utilizes a time delay component to iteratively operation on a precursor pulse, in accordance with one or more embodiments of this disclosure.

FIG. 6A illustrates an example of pulse-width stretching component that utilizes multiple logic gates, in accordance with one or more embodiments of this disclosure.

FIG. 6B illustrates an output continuous pulse signal that results from traversing the pulse-width stretching component shown in FIG. 6A, in accordance with one or more embodiments of this disclosure.

FIG. 7 illustrates an example of matrices R_(α) ₁ _(,k) measured from FPGA α₁ where columns in a matrix correspond to repeated bit measurements for a particular challenge C and rows in a matrix correspond to output bits from different challenges C, in accordance with one or more embodiments of this disclosure.

FIG. 8 illustrates a comparison between two bit sequences resulting from iteratively operating on two input signals by means of respective PUF devices, in accordance with one or more embodiments of this disclosure. The PUF devices are nominally identical.

FIG. 9A illustrates an example of output v(t) from a tent map with switching state s(t), in accordance with one or more embodiments of this disclosure.

FIG. 9B illustrates an example of pulse widths w_(i) from v(t) as a function of i, in accordance with one or more embodiments of this disclosure.

FIG. 10 illustrates an example of a method for providing continuous-time chaos-based conversion of a pulse width to binary bits, in accordance with one or more embodiments of this disclosure.

FIG. 11 illustrates an example of a method for providing a PUF via pulse width chaotic map.

FIG. 12 illustrates another example of a method for providing a PUF via pulse width chaotic map.

FIG. 13 illustrates an example of a method for generating data identifying a fingerprint of a PUF device, in accordance with one or more embodiments of this disclosure.

DETAILED DESCRIPTION

Before the present methods and systems are disclosed and described, it is to be understood that the methods and systems are not limited to specific methods, specific components, or to particular implementations. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting.

As used in the specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Ranges may be expressed herein as from “about” one particular value, and/or to “about” another particular value. When such a range is expressed, another embodiment includes from the one particular value and/or to the other particular value. Similarly, when values are expressed as approximations, by use of the antecedent “about,” it will be understood that the particular value forms another embodiment. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint.

“Optional” or “optionally” means that the subsequently described event or circumstance may or may not occur, and that the description includes instances where said event or circumstance occurs and instances where it does not.

Throughout the description and claims of this specification, the word “comprise” and variations of the word, such as “comprising” and “comprises,” means “including but not limited to,” and is not intended to exclude, for example, other components, integers or steps. “Exemplary” means “an example of” and is not intended to convey an indication of a preferred or ideal embodiment. “Such as” is not used in a restrictive sense, but for explanatory purposes.

Disclosed are components that can be used to perform the disclosed methods and systems. These and other components are disclosed herein, and it is understood that when combinations, subsets, interactions, groups, etc. of these components are disclosed that while specific reference of each various individual and collective combinations and permutation of these may not be explicitly disclosed, each is specifically contemplated and described herein, for all methods and systems. This applies to all aspects of this application including, but not limited to, steps in disclosed methods. Thus, if there are a variety of additional steps that can be performed it is understood that each of these additional steps can be performed with any specific embodiment or combination of embodiments of the disclosed methods.

The present methods and systems may be understood more readily by reference to the following detailed description of preferred embodiments and the examples included therein and to the Figures and their previous and following description.

As will be appreciated by one skilled in the art, the methods and systems may take the form of an entirely hardware embodiment, software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the methods and systems may take the form of a computer program product on a computer-readable storage medium having computer-readable program instructions (e.g., computer software) embodied in the storage medium. More particularly, the present methods and systems may take the form of web-implemented computer software. Any suitable computer-readable storage medium may be utilized including hard disks, CD-ROMs, optical storage devices, or magnetic storage devices.

Embodiments of the methods and systems are described below with reference to block diagrams and flowchart illustrations of methods, systems, apparatuses and computer program products. It will be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and flowchart illustrations, respectively, can be implemented by computer program instructions. These computer program instructions may be loaded onto a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions which execute on the computer or other programmable data processing apparatus create a means for implementing the functions specified in the flowchart block or blocks.

These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including computer-readable instructions for implementing the function specified in the flowchart block or blocks. The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer-implemented process such that the instructions that execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart block or blocks.

Accordingly, blocks of the block diagrams and flowchart illustrations support combinations of means for performing the specified functions, combinations of steps for performing the specified functions and program instruction means for performing the specified functions. It will also be understood that each block of the block diagrams and flowchart illustrations, and combinations of blocks in the block diagrams and flowchart illustrations, can be implemented by special purpose hardware-based computer systems that perform the specified functions or steps, or combinations of special purpose hardware and computer instructions.

As is described in greater details below, the disclosed PUF apparatuses and techniques utilize a relationship between modular functions and chaotic maps, and avoid reliance on clocking signals, to convert a continuous pulse width into a digital sequence of is and Os, where that digital sequence contains unique (device-dependent) bits. The present disclosure relates to a PUF methodology and apparatus using pulse width (time-based) chaotic maps. To operate the pulse width chaotic map PUF, an input precursor or challenge pulse is provided, where a time-to-digital conversion (TDC) process can map the continuous time interval of the pulse width a digital signal (e.g., a set of binary signals or bits) that can be recorded/represented. For example, each time interval can be mapped to a digital signal that can be recorded/represented by an iterative chaotic map (e.g., a tent map, a Bernoulli shift map, a logistic map, or similar). Due to the exponential sensitivity of chaotic maps to both initial conditions (precursor pulse width) and parameter values (established during the manufacturing process), the resulting digital signals contain binary bits that are unique across different physical implementations of the PUF.

A PUF apparatus can include a plurality of logic gates. Each logic gate of the plurality of logic gates can convert each signal (e.g., electronic pulse) to asynchronous digital pulses. Each of the one or more logic gates of the plurality of logic gates can stretch and fold a signal of the plurality of signals in time (e.g., generate a pulse-width corresponding to a respective signal of the plurality of signals received) according to an iterative chaotic map (e.g., a tent map, a Bernoulli shift map, a logistic map). The iterative chaotic map can be used to map stretched and folded signals (e.g., generated pulse-widths) to binary bits. The binary bits can represent a digital word that is unique to the specific physical implementation of that PUF. The result is a PUF (e.g., unclocked, asynchronous, and so forth) system that uses the pulse widths of a time-based chaotic map to measure and report device-dependent information.

In some embodiments, the PUF apparatus 100 can be implemented to operate in any physical system (e.g., a non-simulated process). The precursor pulse 15 can be embodied in a wave propagating in a physical medium. The wave can be, for example, an electromagnetic wave or a pressure wave. The physical medium can in turn be a solid or a fluid that permits the propagation of such a wave. In some configurations, the precursor pulse 15 can be vacuum. The binary sequence s 12 also can be embodied in, for example, a pulse train propagating in the physical medium. The pulse operation component 20 can include components that can receive the precursor pulse 15 and the produce the binary sequence s 12, and can generate the pulse train 25 by implementing chaotic maps, in accordance with aspects described herein. In some embodiments, the pulse train 25 can be embodied in a wave that also can propagate in the physical medium. In other embodiments, the target pulse train 25 can be embodied in an electronic signal propagating in solid-state circuitry.

To mitigate the analog effects and experimental noise, the process described can be recursive. For example, the process can be repeated using an iterative time delay component, such as a delay feedback loop.

Embodiments of the disclosed technologies can be implemented in fully-electronic assemblies, including transistors; CMOS logic gates; field programmable gate-arrays (FPGAs); application-specific integrated circuits (ASICs); a combination of the foregoing; or similar. Assemblies that include FPGAs are highly practical platform for implementing the principles of this disclosure because, amongst other things, FPGAs are reconfigurable and can be implemented/updated using software only.

Other embodiments of the technologies can be implemented in numerous physical systems, such as optical systems, opto-electronic systems, or acoustic systems, where the logic signals can be present in either a physical medium (electromagnetic waves, pressure waves, etc.) or in the electronic elements used to measure such signals. Yet other embodiments of the disclosed technologies can be implemented in hybrid systems that combine electronic logic elements with other physical representations.

FIG. 1 and FIGS. 1A-1E illustrate various aspects of a system for PUFs using chaotic maps that operate on the initial condition provided by a continuous-time width of an pulse. Those skilled in the art will appreciate that present methods may be used in systems that employ both digital and analog equipment. One skilled in the art will appreciate that provided herein is a functional description and that the respective functions can be performed by software, hardware, or a combination of software and hardware. Understanding the distinct roles of noise and determinism on a chaos-based PUF system is an important factor in mitigating their effects. As part of the PUF system, chaos can be realized in a clockless continuous-time chaos-based time-to-digital conversion process. The clockless continuous-time chaos-based time-to-digital conversion system can be constructed using asynchronous logic gates (e.g., field programmable gate arrays (FPGAs)) to form a map operator that outputs an unclocked pulse-train (e.g., digital word) of varying widths. The unclocked pulse-train (e.g., digital word) of varying widths can be used to iterate a chaotic map. The unclocked pulse-train (e.g., digital word) of varying widths can indicate the symbolic dynamics of the chaotic map. The advantages of using a chaotic map are that the resolution is determined by noise in the system and that the digital word contains unique bits that are determined by physical, microscopic manufacturing variations of the device.

FIG. 1 is a schematic block diagram of an example of a TDC apparatus 100 used within the PUF, in accordance with one or more embodiments of this disclosure. The TDC apparatus 100 uses continuous/clockless stretching and folding operations of chaotic maps to measure (with a specified resolution) and/or to convert a continuous time interval (e.g., pulse width) into a set of binary signals that can be stored in digital memory. In other words, the TDC apparatus 100 quantizes the width of continuous/unclocked logic signal 15 (referred to as precursor pulse 15; “high” and “low” signals, for example) as they propagate through the TDC apparatus 100, without the use of a clock. To that end, the TDC apparatus 100 includes a pulse operation component 20 that processes the precursor pulse 15 such that the length of the precursor pulse 15 (e.g., a logic signal) is stretched (e.g., amplified), thresholded (e.g., measured against a reference time interval), and folded (e.g., shortened) back into a usable interval as the precursor signal 15 propagates through the pulse operation component 20. The pulse operation component 20 can iterate such a process essentially continuously.

For example, the precursor pulse 15 is an unclocked logic signal, which can be embodied in a square pulse of w₀. As the precursor pulse 15 propagates through the pulse operation component 20, a pulse width w₀ is stretched by a known multiplicative factor m>1 and the resulting pulse has width mw₀. As such, the pulse operation component 20 can perform a time-stretch operation that amplifies small variations in a pulse's width, where that amplification depends directly on the value of m. Manufacturing variations across different physical implements that are nominally identical will cause this parameter value to be different and hence the amplification to have a different effect in each implementation. Note that for a single iteration of the logic signal through the pulse operation component 20, small variations in the pulse width may still be impractical/impossible to measure. However, by repeating the pulse-width stretching over several iterations k, where the output pulse width is described by w_(k+1)=mw_(k), the microscopic differences between two pulse widths will be amplified and eventually be viewed on a macroscopic scale.

Overall, this stretching process can be efficient for performing time-amplification/stretching, but in order to implement such a process many times essentially continuously, a mechanism is needed that keeps the length of the logic signals bounded (e.g., for many iterations, the length of the logic signal can continue to expand in time-domain and, eventually, the length can be too large for the system's dynamic range). Thus, the pulse operation component 20 also can implement a second mechanism that keeps the signals bounded: a folding operation. Here, the logic signal's pulse width is kept bounded using a structure that performs a modular-like operation on w_(k). This operation takes values of w_(k) that would be amplified beyond a certain point and maps them back into a usable interval without undoing the pulse-width stretching that amplifies small differences. An example of this folding mechanism includes operation that decreases the width mw_(k) by a constant amount c when mw_(k) is larger than a threshold value T (e.g., mw_(k)>T) such that the total operation yields mw_(k)−c. threshold value T in pulse width, Thus, as the pulse is stretching iteratively, it is folded back into an interval of interest to keep its value bounded.

Such combination of pulse-width stretching and pulse-width folding directly corresponds to the operations of chaotic maps. Many types of chaotic maps can be implemented (e.g., tent map, logistic map, etc.). Regardless of the type of map, the pulse operation component 20 can perform iterative continuous operations on the length of the precursor signal 15 (e.g., a logic signal or pulse width) to provide an amplified version of the pulse's length, where this amplification is followed by a folding process and thus the overall mechanism is identical to that of chaotic maps.

To obtain binary representations of the precursor pulse 15, in some embodiments, the pulse widths of such chaotic maps can be partitioned using a threshold or partition. In this partitioning, the pulse widths w_(k) that exit the chaotic map can be assigned respective bit values b_(k) based on the selected partition (threshold). For example, if mw_(k)>T, then b_(k)=1 and if mw_(k)≤T then b_(k)=0. This process can be similar to the measuring the symbolic dynamics of a chaotic map.

Thus, the operations of the TDC apparatus 100 can begin with the precursor pulse 15 having an initial pulse width w₀ that enters into the TDC, the pulse operation component 20 can iterate a chaotic map multiple times. After each iteration k of the map, a pulse width w_(k) is generated and assigned a bit value b_(k). Bits generated in such a fashion can be stored in sequential order in a memory component (such as a storage device or storage circuitry). These bit streams can then be used to recover the original width w₀ using algorithms, post processing, or any other means. The number K of iterations of the map represents the bit depth or resolution of the pulse-width measurement/conversion process. More iterations are equivalent to a higher-resolution TDC, where eventually the noise-floor of the system will cause bits to be unstable. This TDC is a component of the PUF process outlined here.

FIG. 1A is a schematic block diagram of an example of a system 100 for the PUF clockless continuous-time chaos-based time-to-digital conversion, in accordance with one or more embodiments of this disclosure. The system 100 can exploit propagation delays of pulses (electronic pulses, input voltage pulses, etc.) as they transmit through logic gates or other types of logic elements. The system 100 can remain in a stable steady state where, v_(out)=0 V, until an input pulse 101 (e.g., electronic pulses, input voltage pulses, and the like) with a continuous width represented by w₀ is provided/fed as an input to the system 100. The input pulse 101 (e.g., analog signal, a representation of timing between events, and the like) can be provided/fed to a logic OR gate 102. The logic OR gate 102 can break the input pulse 101 into asynchronous logic pulses. Without intending to be bound by theory and/or design, the OR gate 102 enables the system 100 to generate a self-sustaining pulse train after an initial pulse (e.g., the input pulse 101) is provided/fed as an input to the system 100. For example, after an initial pulse (e.g., the input pulse 101) is provided/fed as an input to the system 100, an output (e.g., output voltage v_(out)) can be fed back (e.g., routed, provided, etc.) as an input to the OR gate 102 after a time delay component 106. The time delay component 106 (e.g., delay line) can cause a propagation delay time/period τ.

The OR gate 102 can also continuously provide the input v_(in) to latch component 107 (e.g., flip-flop circuitry, latch circuitry, SR-latch, or similar). The latch component 107 can compare a defined time interval to the width of a signal (e.g., the output of the OR gate 102, v_(in)) generated by a chaotic map component 103. Based on such a comparison, the latch component 107 can generate a binary sequence 12. The chaotic map component 103 can iterate dynamics on a one-dimensional (1D) map. The chaotic map can be, for example, a tent map, a Bernoulli shift map, a logistic map, or similar.

The binary sequence 12 can be stored. The binary sequence 12 can be stored as digital bits (e.g., digital bits of a digital word) generated by the latch component 107. The binary sequence 12 can be used to determine/recover the width of the continuous width signal w₀.

The output of the OR gate 102 can continuously provide the input v_(in) (e.g., input pulses, stretched and folded pulses, a stretched and folded input pulse 101, or the like) to the map operator 103. The OR gate 102 can continuously provide the input v_(in) to the chaotic map component 103 according to an OR gate truth table (not shown).

Results of the iterations implemented by the chaotic map component 103 can be can be pulses of varying widths, represented by w₁ through w_(K), respectively, where K is a natural number. Notably, according to the system 100, neighboring pulses (e.g., electronic pulses, voltage pulses, and the like) do not collide, overlap, or directly interact.

For a chaotic map M, each transformation iteration w_(k+1)=M(w_(k)) of a prior transformation iteration w_(k) can be determined as a gain function g(⋅) applied to the output of a folding function f(⋅) having as an argument w_(k). Namely, w_(k+1)=M(w_(k))=g(f(w_(k))). FIG. 1B is a schematic block diagram that illustrates an example of the chaotic map component 103, in accordance with one or more embodiments of this disclosure. As is illustrated, the map component 103 can include pulse-width folding component 104 that can apply a folding function to an input signal. The chaotic map component 103 also can include a gain component 111 that applies a gain function to an output signal from the folding component 104.

Thus, the pulse-width folding component 104 and the pulse-width gain component 111 can receive the input pulse of defined width w_(k) and generate an output pulse of width W_(k+1)=M(w_(k)) that is stretched in time and gain increased relative to the input pulse. For example, an input pulse having w_(k)=1 μs can be stretched in time to yield w_(k+1)=2 μs. The chaotic map component 103 can stretch and/or fold an input pulse according to any value. The pulse-width folding component 104 and the pulse-width gain component 111 can stretch an input pulse to any value. The pulse-width folding circuitry 104 and the pulse-width gain component 111 permit recording and/or representing chaos in a system, such as the system 100, by providing sufficient conditions for the onset of chaos, for example.

In some configurations, the chaotic map component 103 can be embodied in, or can include, a field programmable gate array (FPGA) or an application-specific integrated circuit (ASIC), or a combination of both. Accordingly, in such configurations, the pulse-width folding component 104 can include circuitry that implements the folding function f(⋅). In turn, the pulse-width stretching component 111 can include circuitry that implements the gain function 9(⋅).

Back to FIG. 1A, to demonstrate chaos in a system (e.g., chaos in the system 100), the output of the map operator 103 (e.g., the output voltage v_(out)) can be provided/fed to a comparator, such as the logic OR gate 102. The output of the map operator 103 (e.g., the output voltage v_(out)) can be compared against a reference signal. If output of the map operator 103 is higher than the reference, the output of the OR gate 102 can be set to 1, otherwise, the output of the OR gate 102 can be set to 0.

As previously described, the output of the map operator 103 (e.g., the output voltage v_(out)) can be fed back (e.g., routed, provided, etc.) as an input to the OR gate 102. The output of the OR gate 102 can continuously provide the input v_(in) to the map operator 103. The output v_(out) (e.g., output voltage) can be fed back (e.g., routed, provided, etc.) as an input to the logic OR gate 102 via the time delay component 106 (e.g., delay line). The output of the system 100 can be shifted by 1 (e.g., shifted to the right) beginning with a least significant bit (LSB) of a digital word after each iteration. The process can be repeated to generate another bit of the digital word (e.g., the next bit of a digital word). The time delay component 106 can include, in some embodiments, a plurality of logic elements, such as an even number of NOT gates.

Although the PUF clockless TDC apparatus 100 shown in FIG. 1A relies on a delay feedback loop to permit the iterative operations on the input pulse 101, the disclosure is not so limited. In some embodiments, as is illustrated in FIG. 1C, the clockless TDC apparatus 100 can include an ingestion component 130 that receives the precursor pulse 15 and N chaotic map components 103(1), 103(2), . . . 103(N) arranged in series. In such an embodiment, to generate the pulse train 25, output signal from a first chaotic map component 103(k) is supplied to a second chaotic map component 103(k+1); here, 1≤k<N. Each chaotic map component 103(k) (k=1, 2, . . . N) generates an output pulse signal by applying a particular chaotic map M, where the output pulse signal has a width w_(k). In addition, each chaotic map component 103(k) (k=1, 2, . . . , N) can generate a bit b_(k) that constitutes the binary sequence s 12 (FIG. 1).

In some embodiments, rather than relying entirely on a single chaotic map component 103 and a delay feedback look (see FIG. 1A) or on a series of chaotic map components 103, the PUF clockless TDC apparatus 100 can include a combination of both multiple chaotic map components and one or several delay feedback loops.

FIG. 1D is a schematic block diagram of an example of latch circuitry 117 used to produce the high and low logic signals for generation of the binary sequence 12, in accordance with one or more embodiments of this disclosure. The latch circuitry can embody, or can constitute, the latch component 107, flip flop circuitry, latch circuitry, SR-latch, and the like. FIG. 1D also illustrates a corresponding timing diagram 122. The latch circuitry 117 can include an AND gate 118 that receives a continuous width (w) pulse signal as an input (e.g., input a). The continuous width (w) pulse signal can be delayed by a delay line (Δ) and the AND gate 118 can received the delayed pulse as another input (e.g., input b). The output of the AND gate 118 (e.g., input a+input b) can correspond to an AND gate truth table. The output of the AND gate 118 (e.g., input a +input b) can be input to a NOR gate 119 that is configured as a set-reset (S-R) latch with a NOR gate 120. The output of the NOR gate 120 can be input to a D-latch of a D-type flip-flop 121. The D-type flip-flop 121 can be clocked by a delayed pulse of the continuous width (w) pulse signal. The timing diagram 122 plots the input a, the input b, the output of the AND gate 118, and the output of the D-type flip-flop 121 (s) in time for cases when the width of the continuous width (w) pulse signal is less than or equal to the delay line (Δ) and cases when the continuous width (w) pulse signal is greater than the delay line (Δ).

The latch circuitry 117 can compare a width of a signal (e.g., the output of the OR gate 102, v_(in)) used to iterate a map operator (e.g., the map operator 103, etc.) to a fixed time and generate a binary output. The latch circuitry 117 can generate the binary output to record (e.g., digitally represent) events occurring in time. For example, the binary output can be stored as digital bits (e.g., digital bits of a digital word, etc.) generated by the latch circuitry 117. The binary output can be used to determine/recover the width of a continuous width pulse signal.

As mentioned, the chaotic map component 103 can be embodied in, or can include, an FPGA or an ASIC, or a combination of both. FIGS. 1E-1F are schematic block diagrams of examples of pulse-width folding component 104 in one of such configurations. As is illustrated in FIG. 1E, the pulse-width folding component 104 can be embodied in pulse-width folding circuitry 104 a that implements a tent folding function f(⋅).

As is illustrated in FIG. 1E, the pulse-width folding circuitry 104 a can include a first AND gate 127 and a second AND gate 128 configured as depicted. At least one input of the AND gate 127 and the AND gate 128 can be an inverted input. The input pulse v_(in) can be provided/fed as an input to the AND gate 128 and an inverted input the AND gate 127. An output of the AND gate 128 can be fed back (e.g., routed, provided, etc. . . . ) via the time delay component 106 and provided as an inverted input to the AND gate 128. The AND gate 128 can provide an output to the time delay component 106 according to an AND gate truth table (not shown). The output of the AND gate 128 can be fed back (e.g., routed, provided, etc.) via the time delay component 106 and provided as an input to the AND gate 127. The AND gate 127 can provide the output v_(out) according to an AND gate truth table (not shown). An input pulse 109 (e.g., input pulse 101) of width w_(k) and an output pulse 110 of width w_(a)=f(w_(k)) can be represented by input v_(in) and output v_(a), respectively. In one example, w_(a)=w_(in) for ≤τ and w_(a)=(τ−w_(in)) for τ<w_(in)≤2τ, where τ represents a propagation delay time/period associated with the time delay component 106 (e.g., delay line). Folding of an input pulse (e.g., the input pulse 101) performed by the pulse-width folding circuitry 104 a can be recorded for visualization purposes, for example. As is illustrated in diagram 140 in FIG. 1E, the pulse-width folding circuitry 104 a permits implementing a tent fold function f(w), where the cusp of the tent fold corresponds to a delay Δ provided by the time delay component 106. The pulse-width folding circuitry 104 a can be used with a pulse width gain function to create a tent map operator.

Embodiments of the pulse-width folding circuitry 104 a are not limited to a particular number of logical components. In some embodiments, the pulse-width folding circuitry 104 a can include more than two logical AND gates.

In turn, FIG. 1F illustrates pulse-width folding circuitry 104 b that can implement a shift folding function f(⋅). The pulse-width folding circuitry 104 b can include an array of logic elements (e.g., logic gates, such as AND gates, NOR gates, OR gates, or similar) and feedback loops (e.g., delay lines) configured to shift fold a signal (e.g., the input pulse(s) 109, input pulse 101, or the like). As is illustrated in diagram 150 in FIG. 1F, the pulse-width folding circuitry 104 b permits implementing a shift fold function f(w), where the shift is performed relative to a delay Δ provided by the time delay component 106.

Other configurations of pulse-width folding circuitry can constitute the pulse-width folding component 104 in instances in which the chaotic map component 103 is embodied in, or includes, an FPGA or an ASIC, or a combination of both. For example, the map component 103 can include pulse-width folding circuitry 104 a that implements the folding function f(⋅).

With further reference to configurations in which the chaotic map component 103 can be embodied in, or can include, an FPGA or an ASIC, or a combination of both, FIG. 1G is a schematic block diagram of analog circuitry 160 that can apply an stretching function to an output signal from the digital circuitry 104 a or 104 b (FIG. 1E). Results of the operations of the analog circuitry 160 are described and illustrated with reference to FIG. 2. In the example arrangement shown in FIG. 1G, the analog circuitry 160 includes first integrator circuitry and second integrator circuity. Such an output signal can be embodied in, for example, a square pulse 170 in time-domain having a defined amplitude and a width w. At a rising edge of the square pulse 170, the first integrator circuitry can begin to integrate such an output signal and can yield a time-dependent output signal 220 having a first defined slope (e.g., m/2). At the falling edge of the square pulse 170, the second integrator circuitry can begin to integrate the square pulse 170 and can yield a time-dependent output signal 230 having a second defined slope (e.g., m). The second defined slope of the second integrator circuitry can be a multiple of the first defined slope of the first integrator.

In such an example arrangement, the analog circuitry 160 shown in FIG. 1G also can include comparator circuitry to determine when the output signal 220 of the first integrator circuitry is less than the output signal 230 of the second integrator circuitry. The comparator circuitry can be powered on at the falling edge of the square pulse 170 and can be powered off when a magnitude of the time-dependent output signal 220 is less than the time-dependent output signal 230. Output signal from the comparator circuitry can span a time interval equal to the width w. The analog circuitry can combine the square pulse 170 with the output signal from the comparator circuitry to yield a pulse signal having a defined width w′ corresponding to γw, where y is the ratio between the second defined slope and the first defined slope. As is illustrated in FIG. 2, in an example configuration in which γ=2, w′ is equal to 2w.

The pulse-width gain component 111 can be embodied in pulse-width gain circuitry that implements the gain function g(⋅). An output v_(a) of the pulse-width folding component 104 can be input to the pulse-width gain circuitry. The pulse-width folding component 104 can be embodied in, for example, either one of the pulse-width folding circuitry 104 a or the pulse-width folding circuitry 104 b (see FIG. 1E). Such an output can be provided as an input signal pulse v_(a) having a width w_(in) to the pulse-width gain circuitry.

As described and illustrated in FIG. 1, FIGS. 1A-1E, and FIGS. 2-6, a this PUF system can produce a chaotic map and the corresponding continuous dynamical states. Thus, this method uses a time-stretching function in conjunction with a time-folding function to encode the pulse width without using the discretized measurement unit associated with a clock.

The generation of binary sequences in accordance with aspects of the technologies described herein can be leveraged to produce physically unclonable functions (PUFs) devices that leverage the sensitivity of chaos in order to fingerprint systems and devices that implement logic operations in an unclocked (asynchronous) fashion. A PUF in accordance with this disclosure can collect device-dependent information from logic signals (e.g., “high” and “low” signals) as they propagate through the PUF device without the use of a clock.

As is illustrated in FIG. 3A, a PUF device 310 includes the sequence generator component 10. As such, the PUF device 310 can amplify (or stretch) the length of logic signals (e.g., pulse width) as the logic signal propagates through the PUF device 310. For example, the device can receive a precursor pulse 305 corresponding to an unclocked logic signal, which is represented as a square pulse. The precursor pulse 305 can be transformed into a challenge signal 305 that is operated on iteratively by the pulse operation component 20. Such iterative operations cause the challenge signal 305 to be extended in time-domain and yield a response pulse 315.

FIG. 3B illustrates an example of another PUF device, in accordance with one or more embodiments of this disclosure.

FIG. 3C illustrates an example of yet another PUF device, in accordance with one or more embodiments of this disclosure.

Each one of the PUF devices illustrated in FIGS. 3A-3C can permit detecting small variations in the parameters of the device as a challenge signal (e.g., a logic signal) iteratively undergoes operations, such as pulse width stretching. As mentioned, the pulse operation component 20 can implement a particular time-dependent chaotic map. Regardless of the type of chaotic map, iterative operations are performed on the length of a logic signal to provide amplified versions of the devices parameters, where this amplification is identical to the entropy generation process of chaotic systems. Indeed, the pulse operation component 20 can implement any mechanism that permit amplifying/stretching and then folding the value of a logic signal inputted in a PUF device in accordance with this disclosure.

Similar to TDC apparatuses, PUF devices disclosed herein can be implemented in any physical system (e.g., a non-simulated process) that is configured to implement stretching and folding operations disclosed herein. For example, fully-electronic examples can include transistors, CMOS logic gates, Field-Programmable-Gate-Arrays (FPGAs), and Application-Specific Integrated Circuits (ASICs). Other physical systems such as optical, opto-electronic, or acoustic could also be used, where the logic signals can be present in either a physical medium (electromagnetic/pressure waves) or in the electronics used to measure such signals (i.e., hybrid systems that combine electronic logic elements with other physical representations). Thus, such implementations of this technologies could range from acoustic to laser-based chaotic PUFs with pulsed signals, both of which would add additional layers of security in different applications.

It is noted that unlike conventional FPGA-based PUFs (e.g., ring oscillator, arbiter, etc.), the functionality of this chaos-based PUF is grounded in principles from chaos and information entropy, thus providing rigor around the fundamental basis of its operation and expected performance. Such formal underpinning of the disclosed technologies can permit supporting the scaled deployment of PUFs over many devices and is beneficial for the eventual acceptance of authentication technologies.

It is also noted that at its core, chaos is an ideal mechanism for amplifying microscopic manufacturing variations that are normally impractical to measure. This amplification allows for simple and efficient characterization of FPGA fingerprints, where additional benefits of our chaos-based PUFs include but are not limited to: (i) lower power compared to PUFs that use clocks or continuous oscillations, (ii) simple implementation protocols that do not require complex programming/wiring or FPGA routing, and (iii) a set of software-defined tuning parameters that allow for flexible operation and performance optimization (e.g., parameters can be used to adjust the responses of the PUF to have a mean bit value of 0.5). Also, chaotic systems are inherently difficult to predict with models due to their entropy generation processes and theoretically impossible to replicate/clone in hardware with identical parameters, which are necessary assurances and layers of security.

FIG. 4A illustrates an example of pulse operation component that utilizes a time delay component to iteratively operate on a precursor pulse, in accordance with one or more embodiments of this disclosure. In this instance, a challenge sequence is used to select the physical implementation (e.g., wiring) of the PUF device. For a set of challenge sequence, one or more response bits are captured and stored, where these response bits are unique to the selected physical implementation (e.g., wiring) and unique across nominally identical devices with the same physical implementation.

FIG. 4B illustrates an example of another pulse operation component that utilizes a time delay component to iteratively operate on a precursor pulse, in accordance with one or more embodiments of this disclosure. In this instance, the response bits are determined using a set of precursor pulses, where the physical implementation of the device is fixed in time.

FIG. 5A illustrates an example of pulse operation component that utilizes multiple time-based chaotic map components assembled in series to iteratively operate on a precursor pulse, in accordance with one or more embodiments of this disclosure. Here, the physical implementation (e.g., wiring) is determined by a set of challenge sequences.

FIG. 5B illustrates an example of another pulse operation component that utilizes a time delay component to iteratively operation on a precursor pulse, in accordance with one or more embodiments of this disclosure. In this instance, the response bits are determined using a set of precursor pulses, where the physical implementation of the device is fixed in time.

FIG. 6A illustrates an example of pulse-width stretching component that utilizes multiple logic gates, in accordance with one or more embodiments of this disclosure. In this case, an even number of NOT gates are used in series to provide a propagation delay to an input pulse of width w. This delay line is then tapped at various location, where comparisons are made in pairing taps using AND gates. The resulting AND gate inputs compensate for odd parity of the NOT gates. The resulting AND gate outputs are fed into an inclusive OR gate.

FIG. 6B illustrates an output continuous pulse signal that results from traversing the pulse-width stretching component shown in FIG. 6A, in accordance with one or more embodiments of this disclosure.

FIG. 7 illustrates an example of matrices R_(α) _(1,) _(k) measured from FPGA α₁ (for many FPGAs α_(i) for i=1, 2, 3, . . . N) for key value k, where columns in a matrix correspond to repeated bit measurements for a particular challenge C and rows in a matrix correspond to output bits from different challenges C, in accordance with one or more embodiments of this disclosure. Bit values are averaged across the columns in each response matrix R_(α) ₁ _(,k) and round the result to 0 or 1. These averaged/rounded values are then compiled into a key matrix M_(α) ₁ _(,k) that is 100×10 rows/columns, where each column corresponds to a different key and each row represents the 100 bits of a given key. It is this key matrix that represents a fingerprint of α₁. From here, there are two remaining steps to testing the PUF: (1) repeat the sampling process using the same compiled code on all FPGAs α_(i), the same challenge sequences for each bit in R_(α) _(i) _(,k), with the same number of measurements to compute M_(α) _(i) _(,k), and (2) compute/compare performance metrics from all α_(i),

A formal or quantitative way to examine PUF metrics is to examine key stability, randomness, and uniqueness can be based on PUF performance metrics, such as normalized Intra-Hamming Distances (Intra-HD), Key Randomness Hamming Distances (Rand-HD) and Inter-Hamming Distances (Inter-HD), respectively. Specifically, the Intra-HD is used to quantify the sensitivity of responses to noise/jitter by calculating the percentage of “bit flips” within response matrices. Rand-HD is then a separate metric used to quantify the correlation and bias of the keys within a given key matrix of a PUF.

FIG. 8 illustrates a comparison between two bit sequences resulting from iteratively operating on two input signals by means of respective PUF devices that use chaotic maps, in accordance with one or more embodiments of this disclosure. The PUF devices are nominally identical with only microscopic parameter variations. In the figure, the two sequences of pulse widths w_(i) across PUF₁ and PUF₂ differ after a certain number of iterations. The same is true for the corresponding digital bits b_(i).

FIG. 9A illustrates an example of output v(t) from a tent map with switching state s(t), in accordance with one or more embodiments of this disclosure.

FIG. 9B illustrates an example of pulse widths w_(i) from v(t) as a function of i, in accordance with one or more embodiments of this disclosure.

FIG. 10, FIG. 11 and FIG. 12 illustrate examples of the techniques that emerge from the principles of this disclosure. While the exemplified methods are presented and described as a series of acts or operations that are performed in a sequence, the disclosure is not limited in that respect. Indeed, any one of the methods disclosed herein is not limited by the order (explicit or implied) of a specific sequence of acts or operations. For example, some acts or operations can occur in a different order than what is illustrated herein. In addition, an act or operation can occur concurrently with another act or operation. Further, in some instances, not all acts or operations may be required to implement a method or group of methods disclosed herein. In some embodiments, the acts or operations disclosed in connection with any of the methods in accordance with this disclosure may be machine-accessible instructions (e.g., machine-readable and/or machine-executable instructions) that can be executed or otherwise implemented by one or many signal processing devices and/or can be stored on machine-readable storage media or storage circuitry.

FIG. 10 is a flowchart of an example method 300 for clockless continuous-time chaos-based time-to-digital conversion. At block 310, a system can receive a continuous pulse signal (e.g., input pulse 101 (FIG. 1A)). The continuous pulse signal can be a single asynchronous digital pulse, for example.

At block 320, the system can iteratively generate multiple output signals. The multiple output signals (e.g., output v_(out) (FIG. 1A)) can be iteratively determined based at least on the continuous pulse signal. For example, each output of the plurality of outputs can be associated with an FPGA of the plurality FPGAs. Each output can be mapped to a region of a time-based chaotic map (e.g., a tent map, a Bernoulli shift map, a logistic map, graph 302, graph 402).

At block 330, the system can generate a digital word d using the multiple output signals. The digital word can include binary numbers. The digital word d can include a defined number N of bits: d=d₀d₁d₂ . . . d_(N−2)d_(N−1), where d_(k) represents a bit (k=0, 1 . . . N−1. Each bit in the digital word d corresponds to an output signal of the multiple output signals. For example, a least significant bit (LSB) of the plurality of bits can correspond to the first bit and a most significant bit (MSB) of the plurality of bits can correspond to a last bit.

FIG. 11 is a flowchart of an example method 400 for processing an input signal by means of a clockless PUF device, in accordance with one or more embodiments of this disclosure. At block 410, the clockless PUF device can receive a continuous pulse signal that causes a challenge digital pulse.

At block 420, the asynchronous digital pulse can be stretched and folded. The asynchronous digital pulse can be provided to a pulse-width folding component (e.g., pulse-width folding circuitry 104 a or pulse-width folding circuitry 104 b (FIG. 1E)) and/or a pulse-width stretching component (e.g., pulse-width gain component 111 (FIG. 1B)).

At block 430, a binary signal can be generated. The binary signal can be a sample from the output of the pulse-width stretching component. For example, the output of the pulse-width stretching component can be compared to a threshold. If a value of the output satisfies or exceeds the threshold, a binary value of 1 can be recorded, and if the value of the output does not satisfy the threshold, a binary value of 0 can be recorded. The binary signal can be a quantized digital representation of the separated pulses (e.g., an analog occurrence, etc.). The binary output can be and/or can be used to determine a bit of a digital word including multiple bits.

At block 440, the operations in block 420 and block 440 can be repeated until a termination criterion is satisfied. For example, the process of stretching and folding the asynchronous pulse and determining a binary output can be repeated using a feedback delay loop until precision of the binary output determination is overwhelmed by noise. Repeating the operations in block 420 and block 440 can generate multiple binary output signals that can represent respective bits of a digital word. The digital word can include any number of bits (e.g., N bits), such that d_(k) represents the digital word, where k=0, 1 . . . N−1. Each bit of the plurality of bits can correspond to determined binary. For example, a LSB of the plurality of bits can correspond to a first determined binary output and an MSB of the plurality of bits can correspond to a last determined binary output.

At block 450, a selection component of the clockless PUF device can select a defined number of digital bits from the multiple binary output signals. The selection component can be embodied in, or can include, for example, the bit selection component shown in FIG. 4A or FIG. 5A. The selected digital bits satisfy a reproducibility criterion. For instance, the selected digital bits can be reproducibly over some measurement or series of measurements over different nominally identical PUF devices.

FIG. 12 is a flowchart of an example of a method 500 for processing an input signal by means of a clockless PUF device, in accordance with one or more embodiments of this disclosure, in accordance with one or more embodiments of this disclosure. At block 510, clockless TDC apparatus can receive a continuous pulse signal having a defined width in time domain. For example, the continuous pulse signal can be the embodied in the precursor pulse signal 15 having width w₀. The precursor pulse 15 can result from converting separate pulses representing the timing between events (e.g., asynchronous photon arrivals) into a single digital pulse.

At block 520, the continuous pulse signal can be folded and stretched by applying a chaotic map (e.g., a tent map, a Bernoulli shift map, a logistic map, or the like), resulting in an output continuous pulse signal. The output continuous pulse signal has a second defined width (e.g., w_(k)). For example, the continuous pulse signal can be provided to and/or propagate through a pulse-width folding component (e.g., pulse-width folding circuitry 104 a, pulse-width folding circuitry 105 b) and/or pulse-width gain component (e.g., pulse-width gain component 111). Such components can be configured within an FPGA, for example.

At block 530, the PUF apparatus can generated a digital bit using the output continuous pulse signal that results from applying the chaotic map. For example, the output continuous pulse signal can be provided to a latch component (e.g., latch circuitry 117, flip-flop circuitry, SR-latch, or similar). See FIG. 1A, for example. The latch component can compare the second defined width of the output continuous pulse signal to a defined amount time and can generate the digital bit. The defined amount of time can be a preset time window or another type of predetermined time, for example. The digital bit can embody, for example, an element of the binary sequence 12. In some embodiments, the digital bit can be stored by the latch component. In some instances, the digital bit in combination with previously generated digital bits can be used to determine/recover the defined width of the continuous pulse signal received at block 510. For example, the latch component can generate the digital bit in Gray code. The Gray code can be processed/converted to binary coding, hexadecimal coding, or any other type of coding representative of the continuous width pulse.

The operations included in block 520 and block 530 can occur essentially simultaneously in configurations in which the chaotic map uses a piecewise operator that the threshold of that operator is used to partition the map. In such configurations, the chaotic map component 103 that implements the chaotic map can output both the next pulse width and the digital bit.

At block 540, operations in blocks 520 and 530 can be repeated until a termination criterion is satisfied. In one example, the use of the latch circuitry to generate/store a digital bit and the process of folding and stretching the continuous width pulse can be repeated using a time delay component until a defined number of iterations is exceeded. In another example, the use of the latch circuitry to generate/store a digital bit and the process of folding and stretching the continuous width pulse can be repeated using a time delay component until precision of the binary output determination is overwhelmed by noise. Repeating the operations in blocks 520 and 530 can generate multiple digital bits (e.g., binary outputs) that can represent respective bits of a digital word.

At block 550, the multiple digital bits can be converted to a digital word d. The digital word d can include, for example, a defined number N of bits: d=d₀d₁d₂ . . . d_(N−2)d_(N−1), where d_(k) represents a bit (k=0, 1 . . . N−1). Each bit of the multiple digital bits can correspond, for example, to a respective bit of the binary sequence s 12. For example, a LSB of the multiple digital bits can correspond to a first determined binary output, and an MSB of the multiple digital bits can correspond to a last determined binary output. The multiple digital bits can be used to determine/recover the defined width (e.g., w₀) of the continuous pulse signal received at block 910. The multiple digital bits may be converted as needed. For example, the latch component (e.g., latch circuitry 117 (FIG. 1D), flip-flop circuitry, SR-latch, or similar) can generate digital bits in Gray code. The Gray code can be processed/converted to binary coding, hexadecimal coding, or any other type of coding representative of the continuous width pulse.

At block 560, a selection component of the clockless PUF device can select a defined number of digital bits from the multiple binary output signals. The selection component can be embodied in, or can include, for example, the bit selection component shown in FIG. 4A or FIG. 5A. The selected digital bits satisfy a reproducibility criterion. For instance, the selected digital bits can be reproducibly over some measurement or series of measurements over different nominally identical PUF devices.

FIG. 13 illustrates an example of a method for generating data identifying a fingerprint of a PUF device, in accordance with one or more embodiments of this disclosure. At block 1310, groups of digital bits from probing a PUF device multiple times using challenge signals and a chaotic map. At block 1320, statistics of the selected group of bits can be generated. At block 1330, a fingerprint of the PUF device can be generated based at least on the statistics. At block 1340, data indicative of the fingerprint can be retained in a database.

While the methods and systems have been described in connection with preferred embodiments and specific examples, it is not intended that the scope be limited to the particular embodiments set forth, as the embodiments herein are intended in all respects to be illustrative rather than restrictive.

As used in this application, the terms “environment,” “system,” “module,” “component,” “architecture,” “interface,” “unit,” and the like are intended to encompass an entity that includes either hardware, software, or a combination of hardware and software. Such an entity can be embodied in, or can include, for example, a signal processing device. In another example, the entity can be embodied in, or can include, an apparatus with a defined functionality provided by optical parts, mechanical parts, and/or electronic circuitry. The terms “environment,” “system,” “engine,” “module,” “component,” “architecture,” “interface,” and “unit” can be utilized interchangeably and can be generically referred to functional elements.

A component can be localized on one processing device or distributed between two or more processing devices. Components can communicate via local and/or remote architectures in accordance, for example, with a signal (either analogic or digital) having one or more data packets (e.g., data from one component interacting with another component in a local processing device, distributed processing devices, and/or across a network with other systems via the signal).

As yet another example, a component can be embodied in or can include an apparatus with a defined functionality provided by mechanical parts operated by electric or electronic circuitry that is controlled by a software application or firmware application executed by a processing device. Such a processing device can be internal or external to the apparatus and can execute at least part of the software or firmware application. Still in another example, a component can be embodied in or can include an apparatus that provides defined functionality through electronic components without mechanical parts. The electronic components can include signal processing devices to execute software or firmware that permits or otherwise facilitates, at least in part, the functionality of the electronic components. For the sake of illustration, an example of such processing device(s) includes an integrated circuit (IC), an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), a discrete gate or transistor logic, discrete hardware components, or any combination thereof designed or otherwise configured (e.g., manufactured) to perform the functions described herein.

In some embodiments, components can communicate via local and/or remote processes in accordance, for example, with a signal (either analog or digital) having one or more data packets (e.g., data from one component interacting with another component in a local system, distributed system, and/or across a network such as a wide area network with other systems via the signal). In addition, or in other embodiments, components can communicate or otherwise be coupled via thermal, mechanical, electrical, and/or electromechanical coupling mechanisms (such as conduits, connectors, combinations thereof, or the like). An interface can include input/output (I/O) components as well as associated processors, applications, and/or other programming components.

Conditional language, such as, among others, “can,” “could,” “might,” or “may,” unless specifically stated otherwise, or otherwise understood within the context as used, is generally intended to convey that certain implementations could include, while other implementations do not include, certain features, elements, and/or operations. Thus, such conditional language generally is not intended to imply that features, elements, and/or operations are in any way required for one or more implementations or that one or more implementations necessarily include logic for deciding, with or without user input or prompting, whether these features, elements, and/or operations are included or are to be performed in any particular implementation.

The flowchart and block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of examples of systems, methods, and computer program products according to various embodiments of the present disclosure. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of instructions, which includes one or more machine-executable or computer-executable instructions for implementing the specified operations. It is noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based devices that perform the specified functions or operations or carry out combinations of special purpose hardware and computer instructions.

Unless otherwise expressly stated, it is in no way intended that any method set forth herein be construed as requiring that its steps be performed in a specific order. Accordingly, where a method claim does not actually recite an order to be followed by its steps or it is not otherwise specifically stated in the claims or descriptions that the steps are to be limited to a specific order, it is in no way intended that an order be inferred, in any respect. This holds for any possible non-express basis for interpretation, including: matters of logic with respect to arrangement of steps or operational flow; plain meaning derived from grammatical organization or punctuation; the number or type of embodiments described in the specification.

What has been described herein in the present specification and annexed drawings includes examples of systems, apparatuses, devices, and techniques for time-to-digital conversion using chaotic maps, without reliance on a clocking signal. It is, of course, not possible to describe every conceivable combination of components and/or methods for purposes of describing the various elements of the disclosure, but it can be recognized that many further combinations and permutations of the disclosed elements are possible. Accordingly, it may be apparent that various modifications can be made to the disclosure without departing from the scope or spirit thereof In addition, or as an alternative, other embodiments of the disclosure may be apparent from consideration of the specification and annexed drawings, and practice of the disclosure as presented herein. It is intended that the examples put forth in the specification and annexed drawings be considered, in all respects, as illustrative and not limiting. Although specific terms are employed herein, they are used in a generic and descriptive sense only and not for purposes of limitation. 

1. A method, comprising: receiving, based on a challenge signal, a second signal having a first pulse that is continuous over time, the first pulse defining a first pulse width, wherein the first pulse width has a first duration; generating, based on the second signal, a third signal having a second pulse that is continuous over time, the second pulse defining a second pulse width, wherein the second pulse width has a second duration that is greater than the first duration; generating, based on the third signal, a fourth signal having a third pulse that is continuous over time, the third pulse defining a third pulse width, wherein the third pulse width has a third duration that is less than a reference time interval; and generating, based on the third signal and the fourth signal, a response signal.
 2. The method of claim 1, wherein the third signal and the fourth signal are based on a map.
 3. The method of claim 2, wherein the map comprises one of a Bernoulli shift map, a tent map, or a logistic map.
 4. The method of claim 2, further comprising: receiving a wiring configuration; and configuring the map on a gate array or an integrated circuit according to the wiring configuration.
 5. The method of claim 4, wherein the wiring configuration is based on the challenge signal.
 6. The method of claim 1, further comprising: generating, based on the second signal, a fifth signal having a fourth pulse that is continuous over time defining a fourth pulse width; and generating, based on the fifth signal, a sixth signal having a fifth pulse that is continuous over time defining a fifth pulse width, wherein the fifth pulse width has a fifth duration that is less than the reference time interval, wherein the response signal is further based on the fifth signal and the sixth signal.
 7. The method of claim 1, further comprising: determining that the third duration is greater than the reference time interval; and decreasing the third duration of the third pulse width of the fourth signal according to a predetermined offset, wherein the generating the response signal is further based on the predetermined offset.
 8. The method of claim 1, further comprising: determining that the second duration is less than the reference time interval; and adjusting, based on the second duration being less than the reference time interval, the third pulse width equivalent to the second pulse width.
 9. The method of claim 1, wherein the second duration is based on a predetermined multiplication factor.
 10. A method, comprising: receiving, based on a challenge signal, a wiring configuration; configuring, based on the wiring configuration, a map; generating, based on the challenge signal and the map, a second signal having a first pulse that is continuous over time defining a first pulse width, wherein the first pulse width has a first duration; generating, based on the second signal, a third signal having a second pulse that is continuous over time defining a second pulse width, wherein the second pulse width has a second duration that is less than a reference time interval; and generating, based on the second signal and the third signal, a response signal.
 11. The method of claim 10, wherein the configuring the map comprises configuring a gate array or an integrated circuit.
 12. The method of claim 10, wherein the map comprises one of a Bernoulli shift map, a tent map, or a logistic map.
 13. An apparatus, comprising: stretching circuitry, based on a wiring configuration and a challenge signal, configured to: configure a map based on the wiring configuration, and generate, based on the challenge signal and the map, a second signal having a first pulse that is continuous over time defining a first pulse width, wherein the first pulse width has a first duration; folding circuitry, based on the map and the second signal, configured to: generate a third signal having a second pulse that is continuous over time defining a second pulse width, wherein the second pulse width has a second duration that is less than a reference time interval; and sampling circuitry configured to, based on the second signal and the third signal, output a response signal.
 14. The apparatus of claim 13, wherein the map is based on a gate array or an integrated circuit configured according to the wiring configuration.
 15. The apparatus of claim 13, wherein the map comprises one of a Bernoulli shift map, a tent map, or a logistic map.
 16. The apparatus of claim 13, wherein the folding circuitry is further configure to generate, based on the challenge signal, a fourth signal having a third pulse that is continuous over time defining a third pulse width, the stretching circuitry is further configured to: generate, based on the fourth signal, a fifth signal having a fourth pulse that is continuous over time defining a fourth pulse width, wherein the fourth pulse width has a fourth duration that is less than the reference time interval, and the response signal is further based on the fourth signal and the fifth signal.
 17. The apparatus of claim 13, wherein the folding circuitry is further configured to: determine that the second duration is greater than the reference time interval; and decrease the second duration of the second pulse width of the third signal according to a predetermined offset.
 18. The apparatus of claim 13, wherein the folding circuitry is further configured to: determine that the first duration is less than the reference time interval; and adjust, based on the second duration being less than the reference time interval, the second pulse width equivalent to the first pulse width. 